Section Summary: 13.1

1. Definitions
   • right-hand rule: Curl the fingers of your right hand around the z-axis with your thumb pointing in the positive z-direction, then open your hand 90 : your fingers point in the positive x-direction; closed, they point in the positive y-direction.
   • coordinate planes: analogous to the coordinate axes in 2-dimensions: the surfaces created when one of the three coordinates is set to zero. Each coordinate plane will contain two of the three coordinate axes.
   • octants: space is now divided into 8 sections, by the walls of the coordinate planes.
   • three-dimensional coordinate system: Once coordinate axes are established, coordinates (x,y,z) indicate a point in space. There is a one-to-one correspondence between space points and these triples.
   • coordinates of a point: an ordered triple of the values (x,y,z) indicating the location of the point in the three-dimensional coordinate system.

2. Theorems
   • Distance formula in three-dimensions: is a simple analogue of the formula in two dimensions. The distance between the points and is given by

     

   • Equation of a Sphere: From the distance formula, we can deduce the equation for the sphere centered on C(h,k,l) of radius r. The sphere is all points equidistance from the center, of distance r, which is indicated by

     

3. Properties/Tricks/Hints/Etc.

   Drawing axes is important:

     
   Figure 1: Top: two legitimate representations; Bottom: sin and sacrilege! Don't do it! It's not physically realizable.
   

4. Summary

   This section serves as an introduction to 3-dimensional coordinate systems and the representation of space, rather than the plane. We see that some formulas extend in quite a natural way.

Section Summary: 13.2

1. Definitions
   • vector: a quantity having both direction and magnitude (length). A two-dimensional vector is an ordered pair a ; A three-dimensional vector is an ordered triple a . The numbers , , and are called components of a.

   • A representation of the vector a is a directed line segment from any point A(x,y) to the point . A particular representation of a is the directed line segment from the origin to the point , called the position vector of the point . Representations in 3-d are defined analogously.

     Vectors will generally be written with an arrow, e.g. , since it's hard to write boldface on paper or on the board.

   • parallel vectors: if two vectors are parallel, then one can be written as a scalar multiple of the other:

     

   • unit vector: a vector whose length is 1.

2. Theorems

   None to speak of.

3. Properties/Tricks/Hints/Etc.
   • Given the points and , the vector a with representation is

     

   • The length of the two-dimensional vector a is

     

     The length of the three-dimensional vector a is

     

   • Addition is defined componentwise, so that a + b is defined as

     

     Subtraction is defined in the obvious way.

     Addition is carried out geometrically by putting the tail of vector b to the head of a and creating the vector from the tail of a to the head of b, creating a parallelogram.

   • Multiplication of a vector by a scalar: If c is a scalar (e.g., a real number) and a , then the vector ca is defined by

     

     and similarly for three-dimensional vectors.

   • Properties of vectors:

     

   • Special vectors (the standard basis vectors, of length 1):

     

     Any vector can be expressed as a sum of the standard unit vectors:

     

   • Vectors can be defined in n-dimensions in entirely analogous ways.

4. Summary

   This section simply introduces us to a quantity, called a vector, which allows us to capture both magnitude and direction. This is useful (for example to indicate wind speed and direction on a weather map), and a set of rules and properties are defined to help us to manipulate these quantities.

   Every vector can be expressed as a sum of special ``basis'' vectors, which are of unit size (length 1)..

Section Summary: 13.3 - The dot product

1. Definitions

   If a and b , then the dot product of a and b is the number given by

   

   Two non-zero vectors a and b are called perpendicular or orthogonal if the angle between them is .

   The direction angles of a non-zero vector a are the angles , , and that the vector makes with the positive x-, y-, and z-axes. The cosines of these angles are called the direction cosines.

   The vector projection of b onto a is

   

   The scalar projection of b onto a is

   

   In particular, the unit vector created from a would be

   

   (the vector with the same direction, but unit length).

2. Theorems

   If is the angle between vectors a and b, then

   

   Corollary:

   

   Hence, a and b are orthogonal (i.e. perpendicular)

3. Properties/Tricks/Hints/Etc.

   Properties of the dot product:

   

4. Summary

   The dot product is a way of combining two vectors to get a scalar (i.e., a number). It effectively measures the shadow that one vector casts on the other, and if the dot product is zero, the two vectors are orthogonal (i.e. perpendicular).

Section Summary: 13.4 - the cross product

1. Definitions

   We create the cross-product to be a sort of ``right-hand rule'' operator. We might start by saying that we want the cross-product to have the following results:

   

   Now assume the usual sorts of distributive properties, so that given a and b , we can compute

   

   or

   

   Note that the cross-product is a vector-product, by contrast with the dot product which produces a number (scalar). Furthermore, the direction of the cross-product is determined by using the right-hand rule.

   This cross-product is only defined for three-dimensional vectors, by contrast with the dot product, which is defined for all vectors in all dimensions.

2. Theorems

   

   where is the angle between a and b. This last property can be interpreted geometrically: the norm (length) of the cross product is the area of the parallelogram constructed using vectors a and b.

   This can be generalized to the parallelpiped, constructed of three vectors, whose volume V is given by

   

   If we discover that V is zero, then the vectors a, b and c must be coplanar (the parallelpiped is at most two-dimensional). Note: by symmetry,

   

   Two non-zero vectors a and b are parallel if and only if

   

3. Properties/Tricks/Hints/Etc.

4. Summary

   Thus the cross-product of two vectors produces a vector which is perpendicular to those two vectors, having magnitude the area of the parallelogram created by the two vectors. We can use this product as a means for determining when two vectors are parallel (they have a zero - vector - cross-product).

Section Summary: 13.5 (part a - lines in space)

1. Definitions

   The vector equation of a line L is

   

   where t is a scalar. Suppose that , , ; since this is true component-wise, we have that

   

   These are the parametric equations of the line L.

   These equations are not unique to a line: any point and vector with orientation along the line will give another set of equations.

   We can solve for t in each of the three parametric equations above to get a set of three equations

   

   called the symmetric equations of L.

   Skew lines are lines that do not intersect and are not parallel. Note: this can't happen in the plane! It only happens in three-space (or higher), that lines can pass like ships in the night....

2. Theorems

   None to speak of.

3. Properties/Tricks/Hints/Etc.

   Lines in 3-space (or higher) can pair up in only one of three ways:

   • parallel
   • intersecting
   • skew (missing each other completely yet not oriented in the same way)

4. Summary

   In the first part of section 13.5, we are introduced to various ways of thinking about lines in space. We meet with several different equations of space lines, and see that lines in space behave a little differently than lines in the plane.

Section Summary: 13.5 (part b - planes in space)

1. Definitions

   The parametric equation of a plane P is

   

   The vector equation of a plane P is

   

   or

   

   Suppose that , , ; then

   

   This is the scalar equation of the plane P with normal vector n.

   These equations are not unique to a plane: any vector and vector normal to the plane will give another set of equations.

   By collecting terms in the scalar equation above we find that

   

   where This is called a linear equation in x, y, and z.

   Two planes are parallel if their normal vectors are parallel.

2. Theorems

   None to speak of.

3. Properties/Tricks/Hints/Etc.

   distinct planes in 3-space (or higher) can pair up in only two ways:

   • parallel
   • intersecting in a line
   The line of intersection can be found by solving both scalar equations of the planes simultaneously.

4. Summary

   In the second part of section 13.5, we are introduced to various ways of thinking about planes in space. We meet with several different equations of planes.

Section Summary: 14.1

1. Definitions

   vector-valued function: a function whose domain is the set of real numbers and whose range is a set of vectors. The components of the vector-valued function are called component functions.

   Often the independent variable will be denoted by t, since it will often be the case that we're dealing with time as the independent variable.

   A space curve C is the plot of points (x,y,z), where

   

   as t varies through an interval I. The equations for the coordinates are called parametric equations of C and t is called a parameter.

   This curve can be considered the path of the tip of a vector-valued function

2. Theorems

   None to speak of.

3. Properties/Tricks/Hints/Etc.

   Many of the ordinary rules of functions pass over directly to vector-valued functions: limits, continuity, etc.

4. Summary

   Vector-valued functions are introduced, and some examples of space curves, which can be considered the paths of the tips of vector-valued functions, are given (e.g. twisted cubics, toroidal spirals, trefoil knots).

   Many of the usual operations of real-valued functions pass directly over to vector-valued functions (e.g. continuity), only on a component-by-component basis.

Section Summary: 14.2

This section simply ``states the obvious'' in some important cases of the usual calculus operations: they will be carried out in the obvious way - component-wise!

1. Definitions

   The derivative of vector-valued function r with respect to parameter t, where , is given by

   

   provided that the component functions are differentiable functions. One might choose to understand as the tangent vector to the space curve of the motion at time t. The second derivative is obtained in the obvious way, by differentiating the derivative function component by component.

   Speed of the point in motion is given by the norm of the vector derivative:

   

   Smooth: a space curve given by the vector function r(t) on an interval I is called smooth if r' is continuous and on I, with the possible exception of the end points.

   One operatives similarly for integrals:

   

   where R is an anti-derivative of r. The usual integration rules apply....

2. Theorems

   The usual rules of differentiation apply: suppose that u and v are differentiable vector-valued functions, c is a scalar, and f is a real-valued function. Then

   

3. Properties/Tricks/Hints/Etc.

   None to speak of.

4. Summary

   The (very good!) news is that the basic operations of differentiation and integration for vector-valued functions are carried out exactly the way we would expect, including the chain rule, the sum rule, and three different versions of the product rule!